Subgroups — C₄₁×#33800 #23810001

Order 10496.

502 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2255	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2244	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2233	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2232	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2228	C₂	1	4	yes	maximal, complemented
8	C₄₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#532	—	2	1	no	—
10	C₂×#125×C₄₁	—	2	1	no	—
11	C₄₁×#500	C₂²	1	1	yes	characteristic
12	C₄₁×#499	—	2	1	no	—
13	C₄₁×#499	—	2	1	no	—
14	C₄₁×#499	C₂²	1	2	yes	—
15	C₄₁×#499	C₂²	1	2	yes	—
16	C₄₁×#493	C₂²	1	1	yes	characteristic
17	C₄₁×#486	—	2	1	no	—
18	C₄₁×#466	—	2	1	no	—
19	C₄₁×#466	C₂²	1	2	yes	—
20	C₄×D₈×C₄₁	—	2	1	no	—
21	C₄×D₈×C₄₁	C₂²	1	2	yes	—
22	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#106×C₄₁	—	2	1	no	—
25	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#367	C₂²	1	4	yes	—
27	C₄₁×#366	C₂²	1	4	yes	—
28	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₄₁×#357	C₂²	1	4	yes	—
30	C₄₁×#357	C₂²	1	4	yes	—
31	C₄₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₄₁×#346	C₂²	1	2	yes	—
33	C₂×#41×C₄₁	—	4	1	no	—
34	C₂×D₈×C₄₁	—	2	1	no	—
35	C₂×D₈×C₄₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₄₁	—	2	1	no	—
39	#127×C₄₁	—	2	1	no	—
40	#125×C₄₁	—	2	2	no	—
41	#125×C₄₁	—	2	2	no	—
42	#125×C₄₁	—	4	1	no	—
43	#125×C₄₁	—	4	1	no	—
44	#124×C₄₁	—	4	1	no	—
45	#121×C₄₁	—	2	2	no	—
46	#121×C₄₁	—	4	1	no	—
47	C₄×D₄×C₄₁	—	2	2	no	—
48	C₄×D₄×C₄₁	—	4	1	no	—
49	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
52	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
53	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
54	C₂×C₁₆×C₄₁	C₂³	1	4	yes	—
55	#107×C₄₁	—	2	1	no	—
56	#107×C₄₁	—	2	1	no	—
57	#107×C₄₁	—	2	1	no	—
58	#107×C₄₁	C₂³	1	2	yes	—
59	#107×C₄₁	C₂³	1	2	yes	—
60	#107×C₄₁	C₂³	1	2	yes	—
61	#106×C₄₁	—	2	2	no	—
62	#106×C₄₁	—	2	2	no	—
63	#102×C₄₁	—	2	2	no	—
64	#102×C₄₁	—	2	2	no	—
65	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
69	#41×C₄₁	—	8	4	no	—
70	C₂×Q₈×C₄₁	—	4	1	no	—
71	C₂×D₄×C₄₁	—	2	2	no	—
72	C₂×D₄×C₄₁	—	4	2	no	—
73	C₂×D₄×C₄₁	—	4	1	no	—
74	C₂×D₄×C₄₁	—	4	1	no	—
75	C₂²×C₄×C₄₁	—	4	1	no	—
76	C₂²×C₄×C₄₁	—	4	1	no	—
77	C₂²×C₄×C₄₁	—	4	2	no	—
78	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
81	D₈×C₄₁	—	2	4	no	—
82	D₈×C₄₁	—	4	2	no	—
83	C₂×C₈×C₄₁	—	2	2	no	—
84	C₂×C₈×C₄₁	—	2	2	no	—
85	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₄₁	—	2	2	no	—
90	C₄⋊C₄×C₄₁	—	2	2	no	—
91	C₄⋊C₄×C₄₁	—	2	2	no	—
92	C₄⋊C₄×C₄₁	—	4	1	no	—
93	C₄⋊C₄×C₄₁	—	4	1	no	—
94	C₄⋊C₄×C₄₁	—	4	2	no	—
95	C₄⋊C₄×C₄₁	—	4	2	no	—
96	C₄⋊C₄×C₄₁	—	4	2	no	—
97	C₄⋊C₄×C₄₁	—	4	2	no	—
98	C₄⋊C₄×C₄₁	—	4	1	no	—
99	C₂²⋊C₄×C₄₁	—	4	2	no	—
100	C₂²⋊C₄×C₄₁	—	4	2	no	—
101	C₄²×C₄₁	D₈	1	1	yes	characteristic
102	C₄²×C₄₁	D₈	1	1	yes	characteristic
103	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
105	C₁₆×C₄₁	—	2	4	no	—
106	C₂³×C₄₁	—	4	2	no	—
107	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
108	Q₈×C₄₁	—	8	2	no	—
109	D₄×C₄₁	—	4	2	no	—
110	D₄×C₄₁	—	4	2	no	—
111	D₄×C₄₁	—	8	2	no	—
112	D₄×C₄₁	—	8	2	no	—
113	D₄×C₄₁	—	8	4	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	2	1	no	—
116	C₂×C₄×C₄₁	—	2	1	no	—
117	C₂×C₄×C₄₁	—	4	1	no	—
118	C₂×C₄×C₄₁	—	4	2	no	—
119	C₂×C₄×C₄₁	—	4	2	no	—
120	C₂×C₄×C₄₁	—	4	2	no	—
121	C₂×C₄×C₄₁	—	8	4	no	—
122	C₂×C₄×C₄₁	—	8	2	no	—
123	C₂×C₄×C₄₁	—	8	2	no	—
124	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₄₁	—	2	1	no	—
131	C₈×C₄₁	—	2	1	no	—
132	C₈×C₄₁	#137	1	2	yes	—
133	C₈×C₄₁	2₋¹⁺⁴	1	2	yes	—
134	#33800	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₄₁	—	2	1	no	—
136	C₂²×C₄₁	—	2	1	no	—
137	C₂²×C₄₁	—	8	2	no	—
138	C₂²×C₄₁	—	8	2	no	—
139	C₂²×C₄₁	—	8	2	no	—
140	C₂²×C₄₁	#509	1	2	yes	—
141	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₄₁	—	2	1	no	—
143	C₄×C₄₁	—	2	1	no	—
144	C₄×C₄₁	—	2	1	no	—
145	C₄×C₄₁	—	2	1	no	—
146	C₄×C₄₁	—	2	1	no	—
147	C₄×C₄₁	—	8	2	no	—
148	C₄×C₄₁	—	8	2	no	—
149	C₄×C₄₁	—	8	2	no	—
150	C₄×C₄₁	#536	1	1	yes	characteristic
151	C₄×C₄₁	C₂×#137	1	1	yes	characteristic
152	#3169	C₂×C₄₁	1	1	yes	characteristic
153	C₂×#499	C₂×C₄₁	1	1	yes	characteristic, complemented
154	#2255	C₂×C₄₁	1	2	yes	complemented
155	#2244	C₂×C₄₁	1	2	yes	complemented
156	#2233	C₂×C₄₁	1	2	yes	complemented
157	#2232	C₂×C₄₁	1	2	yes	complemented
158	#2228	C₂×C₄₁	1	4	yes	complemented
159	#2163	C₂×C₄₁	1	1	yes	characteristic, complemented
160	C₂×C₄₁	—	2	2	no	—
161	C₂×C₄₁	—	16	2	no	—
162	C₂×C₄₁	#3226	1	1	yes	characteristic
163	C₂×C₄₁	C₂×#509	1	1	yes	characteristic
164	C₂×C₄₁	#3414	1	1	yes	characteristic
165	#532	—	2	1	no	—
166	C₂×#125	—	2	1	no	—
167	#500	C₂²×C₄₁	1	1	yes	characteristic
168	#499	—	2	1	no	—
169	#499	—	2	1	no	—
170	#499	C₂²×C₄₁	1	2	yes	—
171	#499	C₂²×C₄₁	1	2	yes	—
172	#493	C₂²×C₄₁	1	1	yes	characteristic
173	#486	—	2	1	no	—
174	#466	—	2	1	no	—
175	#466	C₂²×C₄₁	1	2	yes	—
176	C₄×D₈	—	2	1	no	—
177	C₄×D₈	C₂²×C₄₁	1	2	yes	—
178	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
179	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
180	C₂×#106	—	2	1	no	—
181	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
182	#367	C₂²×C₄₁	1	4	yes	—
183	#366	C₂²×C₄₁	1	4	yes	—
184	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
185	#357	C₂²×C₄₁	1	4	yes	—
186	#357	C₂²×C₄₁	1	4	yes	—
187	C₂²⋊C₁₆	C₂²×C₄₁	1	2	yes	—
188	#346	C₂²×C₄₁	1	2	yes	—
189	C₄₁	#33800	1	1	yes	characteristic, direct product factor, Sylow
190	C₂×#41	—	4	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	C₂³×C₄₁	1	2	yes	—
193	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	1	no	—
196	#127	—	2	1	no	—
197	#125	—	2	2	no	—
198	#125	—	2	2	no	—
199	#125	—	4	1	no	—
200	#125	—	4	1	no	—
201	#124	—	4	1	no	—
202	#121	—	2	2	no	—
203	#121	—	4	1	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	4	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	4	1	no	—
209	C₂×C₄⋊C₄	—	4	1	no	—
210	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
211	C₂×C₁₆	C₂³×C₄₁	1	4	yes	—
212	#107	—	2	1	no	—
213	#107	—	2	1	no	—
214	#107	—	2	1	no	—
215	#107	C₂³×C₄₁	1	2	yes	—
216	#107	C₂³×C₄₁	1	2	yes	—
217	#107	C₂³×C₄₁	1	2	yes	—
218	#106	—	2	2	no	—
219	#106	—	2	2	no	—
220	#102	—	2	2	no	—
221	#102	—	2	2	no	—
222	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
223	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
226	#41	—	8	4	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
236	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
238	D₈	—	2	4	no	—
239	D₈	—	4	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
245	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	D₈×C₄₁	1	1	yes	characteristic
259	C₄²	D₈×C₄₁	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
262	C₁₆	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
265	Q₈	—	8	2	no	—
266	D₄	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	8	2	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
282	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
283	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
284	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
285	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	#137×C₄₁	1	2	yes	—
290	C₈	2₋¹⁺⁴×C₄₁	1	2	yes	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	C₄₁×#509	1	2	yes	—
297	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₄₁×#536	1	1	yes	characteristic
307	C₄	C₂×#137×C₄₁	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	16	2	no	—
310	C₂	C₄₁×#3226	1	1	yes	characteristic
311	C₂	C₂×C₄₁×#509	1	1	yes	characteristic
312	C₂	C₄₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₄₁×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄₁×#499	C₂	semidirect
3. C₄₁×#2255	C₂	semidirect
4. C₄₁×#2244	C₂	semidirect
5. C₄₁×#2233	C₂	semidirect
6. C₄₁×#2232	C₂	semidirect
7. C₄₁×#2228	C₂	semidirect
8. C₄₁×#2163	C₂	semidirect
134. #33800	C₄₁	direct (N × H)
153. C₂×#499	C₂×C₄₁	semidirect
154. #2255	C₂×C₄₁	semidirect
155. #2244	C₂×C₄₁	semidirect
156. #2233	C₂×C₄₁	semidirect
157. #2232	C₂×C₄₁	semidirect
158. #2228	C₂×C₄₁	semidirect
159. #2163	C₂×C₄₁	semidirect
189. C₄₁	#33800	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄₁×#3169	2
C₂×C₄₁×#499	2
C₄₁×#2255	2
C₄₁×#2244	2
C₄₁×#2233	2
C₄₁×#2232	2
C₄₁×#2228	2
C₄₁×#2163	2
#33800	41

Subgroups — C41×#33800 #23810001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33800 #23810001